Given that an inner product on a real vector space $V$ is a function $b : V \times V \rightarrow \mathbb{R}$ satisfying:

$b$ is bilinear (that is, $b$ is linear in the first variable when the second is kept constant and vice versa); and

$b$ is positive definite, that is, $b(v, v) \geq 0$ for all $v \in V,$ and $b(v, v)=0$ if and only if $v=0$,

is it true, given this definition, that all such inner products on a real vector space consist of symmetric bilinear forms? I would be grateful for a proof of this, should it be true.

Many thanks

  • $\begingroup$ Can you prove it for one dimensional vector spaces? $\endgroup$ – Somos May 9 at 10:22

Yes, each inner product on a real vector space is a symmetric bilinear form.


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